Computer won't turn on

There isn't really anything you can do to test a motherboard if the computer will not boot, so that leaves you pretty much screwed. The best thing to do is see if you can purchase the exact type of motherboard to replace it. This way you need only swap out the physical components and not have to backup and reinstall your OS. Newegg.com is a good place to start looking for one.

Also, the problem you have with the computer shutting down while playing games is an overheating issue, like you suspected. The simplest thing to solve this is to clean out the whole case with some canned air and make sure the CPU/graphics card fans are working properly.

Jam it back in, in the dark.